The following is a rough transcript which has not been revised by The Jim Rutt Show or by Barbara Oakley. Please check with us before using any quotations from this transcript. Thank you.
Jim: Today’s guest is Barbara Oakley, professor of industrial and systems engineering at Oakland University in Michigan. Welcome.
Barbara: Well, thank you so much, Jim. It’s a pleasure to be here.
Jim: Yeah, this would be a real interesting currents conversation. Barbara’s got a really unusual background for a professor of engineering. After leaving high school, she didn’t go to college. She enlisted in the US Army, and the Army sent her to study at the University of Washington where she completed a BA in Slavic Languages and Literature. She also received extensive training at the Defense Language Institute. Is that the thing that’s in Monterey?
Barbara: Yep, and it was beautiful.
Jim: Yep, I know people that have gone there, and they say that is one intense program.
Barbara: Yeah, it sure is, but I learned a lot.
Jim: Yep, and we’re going to talk about that, and how that applies to math, interestingly enough.
Jim: Then she went on to serve as a signal officer in Germany and eventually achieved the rank of captain. After she left the Army, she decided to challenge herself. She was basically a liberal artsy-type person, and she wanted to discover if she could retool her mind to study mathematical subjects, and she did. She’s written a book about it, A Mind for Numbers: How to Excel at Math and Science (Even if You Flunked Algebra). Today’s episode will be based on an essay wrote she in Novelists called How I Rewired My Brain to Become Fluent in Math.
Jim: So, a key concept in the essay is fluency, which you repurposed from your experience learning the Russian language. Maybe you could tell us a little bit about that, maybe tell us a few war stories about all that.
Barbara: Oh, I have so many war stories. What has fascinated me over the years is… so, I was terrible at math and science. I mean, I flunked my way through elementary, middle and high school math and science, and I hated… I remember I got called into the Dean of Students when I was in eighth grade because I refused to pay attention, even, in class, and I would just have a stack of books and I’d read a book, and the teacher would come up, grab a book, and I’d just pull another one off my stack. I mean, I was really ostentatiously not going to pay attention.
Barbara: When I was called in, I remember kind of belligerently… I was such a not-too-bright young woman, but I said, “I will never use mathematics. It is a complete waste of time. I don’t want anything to do with it ever.” Then when I enlisted in the Army right out of high school, I went to study Russian, and the Army, they sent me off… I did really well in my studies, and they sent me to get my Bachelors degree in Slavic Languages and Literature. Then, in their great wisdom, they made me a signal officer, which meant that I was in charge of cable systems, radio switchboards, and so forth. I didn’t even know what a volt was. What’s an amp? Why does a circuit have to go out and then come back? Why can’t it just go out? I mean, I didn’t understand any of this stuff, but what did start to become clear is, there’s something to this math business, and technology is actually quite useful, and in fact, it’s really very interesting.
Barbara: I also found that, guess what, there was just not a big call for people whose sole professional expertise is Slavic languages and literature. So, when I decided to go ahead and complete my service and get out of the Army at age 26, I thought, “Well, nobody wants to hire me. I’m supposed to be kind of adventurous and open to learning new things. Why don’t I see if I can change my brain?”
Barbara: What really surprised me was, the techniques I had used to learn language… which, really, very good techniques, because the Defense Language Institute brings in the top experts from around the world, and they had put together programs for learning languages that were outstanding, and they had to because they’re in World War II; you better learn a language fast when you did have to learn one.
Barbara: But a lot of what they emphasized, as I discovered… as I now realized… is that they were emphasizing procedural fluency. So, you didn’t just use… there’s actually two ways that you learn things. You learn through the hippocampus that declarative system; that’s the step by step by step. I learn this, that means this, and then I do this, or I connect the word pato, which means duck in Spanish, with a paddling duck in a little pot. So, those kinds of things are very straightforward to learn, but procedural fluency is how you learn patterns and how you gain automaticity, and you only do that through lots of practice and varied interleaved practice.
Barbara: So, when I start to study math and science as an adult, I thought, “Well, the best way I’ve handled my things is to use this kind of procedural fluency, really get these ideas and interleave them,” and it turned out that approach worked great, and that’s not the approach, surprisingly, that is used in teaching math and science in many of today’s K-12 educators’ approaches to teaching in math and science. So, sometimes I’m a little taken aback, in that it almost seems like this system we have in this country for teaching math and science is crippling students’ abilities to learn in those subject areas.
Jim: Yeah, that called that out quite specifically, huh?
Jim: My own history was a little different. I’ve always loved math and science, and even though I came from… my father dropped out of high school after ninth grade and my mother left home when she was 14, and so I didn’t really have an academic kind of background; grew up in a neighborhood where half the adults were high school dropouts… but I nonetheless always loved math and science. In fact, I still recall and am proud of the fact that I got 100% on every test and quiz in Algebra I for the whole year.
Barbara: Wow, wow.
Jim: Yeah, so I was a math nerd, right? Just loved it for whatever reason. However, I think part of that is that I was lucky in my timing. When I came up, there was a lot of rote memory and learning procedures, how to do long division, even how to calculate a square root, how to solve a quadratic equation. They just pound the hell out of it into you.
Jim: However, there was an attempt when I was about in sixth grade to introduce something called the new math, which was set theoretic, and all the parents rebelled because they said, “I can’t help my kids with the homework. What is this shit?” So we got a little bit of that in our curriculum, but not a lot, so in some sense, I feel like I got the optimal amount, which was some, as you call it, understanding-centered teaching methods, but also tremendous amount of procedural and rote learning, which I believe was what gave me a quite solid math and generally quantitative background.
Jim: Maybe you could dig in a little bit into this so-called understanding-centered teaching method that seems to have come back after my day and become more central in how math is taught today.
Barbara: Well, I think that the challenge is that educators often believe that the only way you really learn something is if you learn it declaratively. That means you can explain it, so it’s step by step by step; you can explain what’s going on. But when you learn something procedurally, you know it really, really well, but you can’t necessarily explain it. That’s why it’s so difficult to… if you know how to solve Rubik’s Cubes, you can’t really explain it very well, or if I asked you to explain how you tie your shoelace… I mean, you could do it in a second, but explaining how you do it? That is really hard because it’s procedural.
Barbara: So, I think that when… you’re right. See, it’s not all bad to think in terms of understanding. That’s actually great. But I remember, I once had a student come up to me, and he waved this test he’d flunked, and I’d had red ink that I put everywhere because it was just terrible. He waved it at me and he said, “I just don’t understand how I could have flunked this test, because I understood what you said when you said it in class.” It’s become so all-encompassing that all you need is this sort of magic feeling of understanding, that that’s enough, that you’ve really learned something, but you really only learn something when you put those sets of links or make those connections between neurons in longterm memory, in the neo-cortex. Just understanding something does not… temporarily, you kind of caught it, but if you don’t practice with it, if you don’t really cement those links, you don’t have it, so you’re just fooling yourself.
Barbara: I’m sure you know the Dunning-Kruger syndrome of, you think you know something, and you don’t know it at all, so in some sense, you’re so stupid that you don’t even realize how stupid you are.
Jim: Yeah. Funny, I have Dunning-Kruger in my notes. I was going to bring that forward. A fine example is comparing U.S. high school students with East Asian high school students.
Barbara: Oh yeah.
Jim: The American high school students think they know math at a pretty high level. Their level of confidence in knowing is high, but their actual ability to perform is low. Well, in East Asia, it’s the opposite. The East Asians are relatively unconfident about their knowledge of math, but they can perform at a high level. So, that’s kind of really interesting that our American way of teaching, which you call understanding-centered, has clearly resulted in Dunning-Kruger syndrome with respect to actual ability to do math.
Barbara: That’s right. I think a problem is… so, there are several problems. What educators will do… if you bring up, for example, that the PISA tests, that Asians often do far, far better in PISA tests, and in indeed, you bring someone from China in eighth grade to do eighth grade U.S. math, and it’s a joke to them what we’re covering. And so, the response by educators, unfortunately, has been just to demean the PISA tests. “Oh no, that’s just a stupid test. It’s not really testing the full abilities,” which, actually, it’s testing some important abilities, and if you’re going down on that slope of PISA tests, that’s not a good sign.
Barbara: But the other thing that they do that is, to my mind, even worse is that they conflate challenges in, for example, the Chinese educational system with challenges in the U.S. system, and what I mean by that is educators will often say, “Well, we don’t want our students having this rote memorization approach to learning in math,” and we don’t… I mean, to some extent… but the challenge is, that the Chinese educational system also does something really, really different. I mean, it idolizes teaches to such an extent that you never want to object or ask questions, or do any of these kinds of things.
Barbara: So, it’s a cultural approach to learning that can, in some sense, suppress creativity, but it has nothing to do with the fact that you’ve practiced something really well and you’ve gotten really good at it. As you do in math, so you need to do when you’re learning how to play the piano, or learning a new language and so forth; you need that practice, and it certainly doesn’t suppress creativity if you’ve practiced a lot with a piano or with learning a language, or learning how to ride a bicycle, but somehow… I know that when I expressed in a New York Times op-ed that practice is important in learning math, and there’s plenty of evidence about that in research, which should be an utterly benign statement, but you would have that I’d just said, “Well, we should burn down all elementary schools in this country,” because math educators are like, “Practice? Oh no, that’s just evil. You shouldn’t be doing that kind of thing, because that’s what kills students’ love of the material,” and it’s like, “No, it’s not. It’s actually practice that helps students master it and grow to love it.”
Barbara: In fact, if you look at hiring committees that I’ve been on, virtually every one of thousands of applicants are from countries that have the methods of learning math that are dismissed by U.S. educators. What is going on here that all of our engineering applicants are actually from these countries who have, so to speak, the wrong approach to learning? Clearly, it’s the right approach. It’s helped them.
Barbara: So, I think the educators need to perk up and look at what’s going on instead of sort of circling the wagons and thinking defensively that their approach has to be right.
Jim: Yeah, it’s unfortunate that education departments, even in the most elite universities, are often 50 years behind the times with respect to things like cognitive science, or even, as you pointed out, plain old results. It’s an extraordinarily conservative, in the sense of looking backward in time… well, I could tell some stories of there, but I probably shouldn’t, so I won’t.
Barbara: Oh, no, I want to hear them!
Jim: Oh, goodness. Well, I’m going to tell you one story. I was helping an elite university design a Ph.D. program in cognitive science, cross-disciplinary, where essentially it would be a… a Ph.D. is an applied cognitive science across… we got the Dean of the Business School to sign in, the Dean of the Medical School to sign up, the Dean of Engineering, and of course, we had the Dean of Arts and Sciences where psychology was, which had a very strong cognitive science department. Everybody was on board, and along the way, we also decided to go talk to the Education School. Guess who told us they weren’t interested at all? The Education School.
Barbara: I remember once pursuing the Dean of the School of Education down and around a hallway because she refused to sign off that she was supporting a grant proposal that would be about practice as a method of helping students learn math. So, her way of refusing to do it was not to just outright refuse, but to always just not be there, so she couldn’t sign off on things. I remember seeing her and she saw me, and she took off down the hallway, and I was just like, “This is just so weird.”
Barbara: There is some visionary work being done in education, and cognitive psychology has done some fantastic research that I based my work on. I certainly don’t want to dismiss everything, but I do have to laugh, because lately, there’s this… well, there’s long been this flood of, “Oh, online learning is terrible. It’s just awful,” and now, there’s this huge flood of, “Online learning is a terrible thing. Our students are losing this year because it’s online, and online is clearly worse than in-class.” Yet, I look at some of the biggest proponents, so-called experts who are saying this and shouting it from the rooftops, and if you look at their online courses, they’re absolutely awful. I mean, they’re just awful.
Barbara: And so, you can see that one reason they don’t like online learning is because A, they can’t trap students in classrooms and make them pay attention because they’re trapped there, and yet that’s what good teaching is. So, you’ll watch some of their teaching, and they have no illustrations; they’ll just get up and talk and use their voice, and of course, people don’t like that kind of thing. So, it does make me laugh, because I think that as terrible as it is, COVID is getting some educators to realize that, “Wait a minute, there is at least a little bit of something good going on with online learning that we really can teach in this way.”
Barbara: My favorite athlete of all time is Julius Yego, and he wanted to throw the javelin. He was from Kenya, and of course, there was no javelin-throwing coaches, so he just watched YouTube and taught himself how to throw the javelin, and he, 99.9% just by watching YouTube and going out and training on his own, he won the World Championship in javelin. I’m just like, people can learn so much online, and I think that some of visionaries are becoming more and more aware of that.
Jim: Yeah, certainly, this is an opportunity for society, this COVID mess, in a lot of ways, and I’ve had episodes on my shows about that. I think one of the things is it will at least start to sort out the crap from the good in online learning.
Jim: In math, I know when I run into somebody that seems to need to learn some math, and has a hole in their math background, I’ll often point them to the Khan Academy stuff. What do you think about that?
Barbara: Oh, I adore Khan Academy, and in fact, our daughter is in her Masters program in Statistics now, our younger daughter, and she swears that she only got through her first years of college math because of the support of Khan Academy. Again, it does make me laugh, because you’ll look at some of the reform educators and they’ll be like, “He actually explains this stuff! Shame on him! Students should figure it out on their own.” It’s like, that maybe works when you’re eight years old, but when you’re nearly 20 and you’re studying advanced calculus, it’s just not so helpful to try and figure out everything out on your own. Not everybody is a Newton, so it’s just kind of a sticky wicket there sometimes.
Jim: Yeah, indeed. One of my friends, Zach Stein, has written a fair amount about innovative education. He calls it the need for teacherly authority. The idea that the kids can find their own way through the underbrush is just not realistic, at least not in any reasonable period of time, and so the teacher has to point the way, essentially.
Barbara: There is so much evidence from evolutionary theory that supports precisely that. If you look at Dave Geary’s work, there’s plenty of evidence that the more common… there’s two ways of categorizing knowledge, and I call it the easy stuff and the hard stuff, but the more formal term is biologically primary and biologically secondary material.
Barbara: So, biologically primary material is stuff like being able to recognize a face or being able to speak a native language, and those are really, really difficult tasks, but we do it easily. We don’t even need to be trained. I mean, boy, would universities love it if you had to take an advanced course in recognizing faces or something, and for artificial intelligence to do it, it’s taken decades and decades.
Barbara: But the more advanced kind of material are things like reading, writing, and doing mathematics; none of these things were what we had done from an evolutionary perspective, and to do those kinds of things, you have to actually kind of push wiring and rewire your brain, which is why it’s important to learn these kinds of things when you’re younger, because it helps that rewiring to take place if you’re young while it’s pushing things aside.
Barbara: The more biologically secondary the material… in other words, the harder the material… the more that explicit instruction mixed with episodes of active learning becomes important. There’s one paper by Freeman, and it did make me laugh… it’s like, Active Learning… I’m paraphrasing on the title, but it’s Active Learning is Super Important in STEM Education, and it’s a very well-done paper, and it’s a large meta analysis that, in essence, concludes that active learning is the be all and end all, that without some active learning in the classroom, the students are not going to be as successful.
Barbara: But the implication of that paper is that you should always spend all your time doing active learning. That’s not exactly what they say, but if you look at the title and sort of the implication, what people walk away from that paper with is, well, you should just have students be figuring things out and doing things actively all the time. Yet, there’s another paper, another massive meta analysis published eight years before that, that says that active learning really doesn’t do squat. It’s really problematic, discovery-based, experiential, all these kinds of things. They’re not good methods. They produce failures.
Barbara: So, who’s right? Is it Freeman or Kirschner? Which paper is correct? The answer is both papers are correct; it’s just that… it’s the interspersing of explicit instruction with bits of active learning, with bits of explicit instruction, back and forth… that’s what’s really, really helpful for students. This mixture is called direct instruction and it’s often denigrated by educators because they don’t understand that it’s a mix of active learning with explicit instruction. They think it’s just talk and chalk through an entire… but it’s not, and it’s been very well researched as one of the best ways for students to learn.
Barbara: It’s kind of funny, in that we do understand a lot about how people learn effectively. It’s just that that business of, when an expert gets in mind their idea of what something should be, they can be utterly inflexible about anything different, any different approaches, because it could affect their career. Even long ago, Santiago Ramon y Cajal, the wonderful founder of… he’s called the father of modern neuroscience… he said one of his greatest attributes was that he was flexible, that he was no genius, that he wasn’t that smart, because he didn’t grow up being really smart, and that meant he was used to making mistakes and correcting his mistakes. He said the geniuses he worked with, they weren’t used to correcting their mistakes, and what they would do instead was they would just kind of find ways to justify why they must have been right after all.
Barbara: I think we see that going on with, not only educational gurus, but gurus in pretty much any kind of enterprise.
Jim: Yeah, justification is a core of human cognitive capability. In fact, we’ve had Gregg Henriques on a couple of times on the show, who’s a professor at James Madison University, and his theory of human cognition is called justification theory, that that’s what humans do. That’s a little overstated, but a big part of his broader tree of knowledge theory is justification theory, and that humans, unlike other animals… well, the new capabilities that we added since the chimpanzee is this capability to justify our own actions… very strong, unfortunately.
Barbara: Yeah. That is really fascinating, but what I find interesting… so, when I’m teaching engineering students, I’ve gotten these cocky… they’re usually guys, and they’re totally cocky, and the thing is you want to say, “Don’t be so cocky about stuff,” especially when they’ve got Dunning-Kruger syndrome. But at the same time, a lot of engineering successes have occurred because of this cockiness. You look at Isambard Kingdom Brunel and his brilliant bridges and tunnels and things, and it’s like, “This is all impossible stuff. You can’t be doing this.” It was his sheer cockiness that got him off the ground, so to speak, or literally, whereas if you were a little bit more willing to accept your own flaws and so forth, but almost too much so, then you can end up in trouble because you’re indecisive and it’s hard to get things done.
Barbara: I’ve worked with people, and I’m sure you have, too, where they just ask you… they’re incapable of independently making decisions, and that just can drive you crazy, too.
Jim: Yeah, very similar to business; in my business career, especially entrepreneurs… I did a number of startups and I’ve helped other startups. Truthfully, if you’re not a little bit overconfident, you probably shouldn’t be in the startup game, right?
Barbara: Right. Right.
Jim: You have to believe probably more than what a cold-blooded analysis would say. On the other hand, you can’t be too crazy. You can’t be disconnected from reality, so it’s this… I call it an optimistic bias, but grounded in reality, is probably the right place to be, something like that.
Barbara: Yeah. I think that’s a good way of putting it. I’ve heard that one of the biggest challenges in training diplomats, though, is that they tend to be overly cocky. They’re really convinced that their cultural approach is the right way to approach things, and that if they go to another country that does things differently, well, they’re wrong. One of the most difficult challenges in training high-level diplomats is to get them to realize that there are other ways of thinking about things, and your way is not necessarily the right way.
Jim: Yeah, but certainly in those kinds of cases where you’re talking about cross-cultural, that’s very important to have a certain amount of modesty.
Barbara: Right. Oh yeah.
Jim: Before we move on to the next topic, I was just going to mention, you mentioned direct instruction, which indeed is, by every measure, a very successful way to teach. Mention that at School of Education and they’ll get the guns and the flaming torches out. That’s like the worst thing you can say at a school of education.
Barbara: Isn’t it funny? It’s just a shame. Part of it, I think, is that… when you really look at direct instruction, it’s that mixture of explanation plus active learning, and that just makes common sense. The students are very much involved in what they’re doing. But it’s this weird thing, because I’ll talk to some people and they’ll say, it’s just not true that instructivist approaches, which means basically student-centered learning, that students are put in charge of their learning… that it’s just not true that teachers will just dump students off to do their own learning. Constructivism actually entails explaining and then having students do things, so it’s almost like they’ll just put a different set of terminology and we’re actually talking about the same things as being optimal for students.
Barbara: But what I have come to discover is that very high-level educators who are affiliated with really good schools… it’s absolutely true, they will mix explicit instruction with active learning, and so their form of constructivism is a very powerful one, but they are often completely unaware of what is going on in a typical inner urban school, for example, or a rural school, or where when these constructivist approaches are promulgated, it often can end up as, okay, you guys, you do your stuff, and I’m going to answer my emails and check what I’m working on, because your job is to teach yourself.
Barbara: So, it’s almost like there’s this disconnect of what high-level educators actually know about what’s really occurring in the classroom in many of these sort of extended rural or urban schools that are not high-level and not wealthy and so forth.
Jim: Interesting. I know, for example, some charter schools that have focused on direct instructions in the inner city have done remarkably well taking the same students.
Barbara: Absolutely, and what’s funny is… let’s see, if you read Eva Moskowitz’s The Education of Eva Moskowitz, and she talks about starting up a series of charter schools in New York City, and she was going up against the head of the teachers’ union who was also starting sort of a equivalent charter school. So, it was going to be, the proof is right there; they’re going mano a mano at each other, and of course, Eva’s schools came out stellar, and the other school… I believe they had to shut that school system down because it was just doing so terribly, and then, of course, the teachers’ union was like, “Ah, somebody else took over and they weren’t doing it right,” and justifying things.
Barbara: I do think that charter schools, they’re not perfect. There’s no system that is perfect, but to have a monolithic lock on how children are taught is a recipe for some pretty poor approaches for students.
Jim: Yeah, unfortunately we’re seeing it in the results.
Jim: Let’s go on to the next topic. You talk about the relationship between learning math and learning a sport. You talk a little bit about golf; you may have some other examples. Maybe you can dig into that a little bit.
Barbara: Okay, so when you’re learning something, you can learn through the declarative system. What’s happening is your working memory, which is kind of like what you can hold temporarily in mind, is sending information through the hippocampus into longterm memory, and the hippocampus kind of massages things. Eventually, you can get to learn it so you don’t even have to use the hippocampus; you can kind of just draw something from longterm memory. This is a very flexible system of learning, and it also allows you to… I mean, you can learn it quickly, because for example, if you are learning a verb conjugation pattern in Spanish, then you don’t have to just listen to all these hundreds and hundreds of sentences and see if you can figure out what the conjugation pattern is; you are directly told what that pattern is.
Barbara: So it’s easy to learn, but it’s really slow to use. If I’m trying to say something in Spanish, I’m like, “Now, is that hablo, hablas, hable… no, hablando… which one is it?” So it’s slow to use; fast to learn, but slow to use. I can get around to getting the right answer, but not with a speedy, fluent proficiency.
Barbara: However, if I learn some facts through my procedural system… the procedural system, it largely encompasses the basal ganglia. You are completely unaware of how that system learns. It’s like you’re trying to hit a ball; well, you tell yourself, “Okay, hit the ball,” and somehow magically inside your system, that procedural system, it figures out through enough practice how to hit the ball, or how to type on the keyboard, or how to drive the car.
Barbara: The thing is, it’s very slow to learn, but it’s really fast to use. So, when I’m typing on the keyboard, don’t even ask me the second finger from the end, which keys does that type? I couldn’t tell you, but I can type really speedily. But it’s inflexible; if you change the keyboard keys, it would take me a long time to adjust to that.
Barbara: Now when you are learning something like a language, you are learning both through your declarative system… you’re setting links there… and that’s really that… it’s fast to put those links there. You’re also learning through the procedural system, and at least you hope so. What can happen with people is they can be learning a language, and they’re learning everything declaratively, and then they actually meet a person from that country… say, Chile… and that person talks to them, and they know the words, they know what they want to say, and nothing comes out because there’s no procedural fluency. You have not set those sets of links through the procedural system to draw on, so the result is you stand there with your mouth open and you can’t say a word, of if you do talk, it’s very halting. So, this is why it’s important to learn both declaratively and procedurally so that you’re not just speaking, but you’re speaking fluently and easily without having to think about it.
Barbara: This is also the case with math. You want to have it so that when you look at a problem, you intuitively have a sense of how to solve that problem, and that only comes through lots of practice, and interleaved practice; that means mixing up somewhat similar types of problems so you know that, for example, oh, you’ve got to use the geometric distribution here, as opposed to binomial approaches there. You know which one to select based on the fact that you’ve practiced a lot.
Barbara: So, procedural fluency develops through practice, a lot of interleaved practice, and both are very important in learning, for example, mathematics. So if you have been educated in a way that only emphasizes declarative learning, you can figure out things, but it’s a lot slower, it’s not very intuitive, and it kind of cripples your ability to easily, so to speak, speak in that language, because you’re just not fluent in it. You haven’t practiced enough to develop that fluency.
Jim: That’s beautiful. I’ll give you an example, something I discovered in my own life that fits your model perfectly. There was a period where I was doing tutoring for pre-meds in chemistry, and this was sometimes after I was in college… a little side hustle, as they say… and these, of course, being pre-meds, they are Type A, good nerdy little students, et cetera, but these were ones that were struggling in Freshman Chemistry, and I found again and again and again… and I basically was probing on their very basic math skills. I found that more than 50%, their fundamental problem… you’ll love this, as somebody who has mastered this stuff… the failure was in the most basic manipulations around fractions, the stuff you learned in fifth grade, basically.
Jim: I found one, in particular, which was, here they are, Type A pre-meds at a pretty elite college, who could not fluently… and I love your word, fluently… manipulate, or let’s say more specifically, add fractions with different denominators, right? When you’re doing dimensional reduction, dimensional analysis in chemistry, and solving rate equations and things like that, you’re constantly manipulating fractions. That one little, literally, fifth grade or sixth grade… I guess it got to fifth grade, adding fractions with different denominators skill… they understood the theory. Could you explain how to do it? Yeah, sort of, maybe, but could they do it fluently when you had to do it in production, when you’re doing your homework or an exam? Absolutely not.
Jim: So, I would find these simple procedural holes that they had, and it was almost always in fractions, hilariously. People understood decimals fairly well, but they did not understand fractions. I just drilled the hell out of them until they had, at their fingertips, unconscious procedural memory on how to do the relatively simple mechanism to add two fractions with different denominator.
Barbara: You are right at the point, and I think even underneath this problem with fractions, is the fact that multiplication tables are largely discouraged. Educators discourage learning the multiplication tables by heart. You can’t manipulate fractions very easily if you do not know the multiplication tables. So, I do see that… for example, the same as you, I would see people who they’d look at something like one over, one over K, and they’d look at that and they’d be like, “What is that? How do I handle that?” It’s a very simple… I mean, if you practiced, you’d look at that and go, “That’s just K.”
Jim: Yeah, just flip it over, right? We know that! All of us math nerds have gotten that beaten into our heads over the years by constant practice.
Barbara: Right, but they didn’t have that constant practice, because practice is kind of bad. I mean, the simplest things will goof students up, and part of it, I think… there’s this self-serving tendency by big technology to promote this idea that you can always just look it up. Sometimes I’m like, “Are you looking at who is saying this?” I mean, it’s a math calculation website. Of course, they want you to look it up! They don’t want you to be able to handle these things mentally.
Barbara: Of course, you need to have calculators to do the advanced sorts of calculations, but by golly, you’ve got to have those basics, and if you don’t, the simplest things are going to trip you up, and I think that’s what we see time and time again now, as you noticed even in your tutoring career.
Jim: Yeah, another thing I noticed in business was that folks who did not have, let’s say, multiplication tables or exponentials at their fingertips… their ability to roughly estimate an answer intuitively was completely lacking.
Barbara: Oh yeah.
Jim: They couldn’t even get it within an order of magnitude because they just didn’t have the upholstery. I sometimes use the analogy that learning only the concepts and not actually the details or the facts is kind of like having a architect-designed house with no furniture.
Barbara: That is really a good analogy.
Jim: It was really scary. I mean, a person who has decent intuitive math skills can estimate, and in fact, one of my best friends, who is the smartest person in my class at MIT, probably… he’d always say, “God damn it, there’s never going to be a manned space mission that’s successful because they’re all using calculators now and not slide rules.”
Jim: He predicted the shuttle disasters in advance based on that. He said the shuttle is the first device that was created using calculators and computers, not slide rules, and if you use a slide rule, you have to have intuitive feeling for orders of magnitude, and you’ll never make orders of magnitude-type errors.
Barbara: Yep, that’s so true. The thing that gets me is that constructivist approaches emphasize that you have to build your knowledge internally, and that’s so true that it’s so important, and then it all gets thrown out… the baby gets thrown out with the bath water when it comes to, yeah, but you don’t need to internalize anything to do with mathematics because you can always just look it up. It’s like, no, then you don’t get a feel for the patterns. You can’t make these kinds of very, very important estimations.
Jim: Yep, I love your word, fluency. I never put fluency together with this kind of intuition, but I think that it does a remarkable job of getting the sense of what needs to be done here with math education in the same way that the big difference… yeah, I took my three years of high school Spanish and I can say dos cervezas when I go to Mexico, but that’s about it. On the other hand, I know people always say, “If you really want to learn a language, have a romantic relationship with somebody from that country who doesn’t speak English. You’ll figure it out and you’ll become fluent.” That’s a very, very different thing than just understanding the theory.
Jim: Well, I really want to thank you for this episode. This has been really wonderful, and we talked about pre-game. I’m going to have you back to talk about your book that’s coming out this summer, Uncommon Sense Teaching: Practical Insights in Brain Science to Help Students Learn. So, thanks for this great episode, and I look forward to having you back.
Barbara: Well, thank you, Jim. I look forward to being back.
Production services and audio editing by Jared Janes Consulting, Music by Tom Muller at modernspacemusic.com.